The figure shows line segment PQ and a circle with radius 1 and center (5, 2) in the xy-plane. Which of the following values could be the distance between a point on line segment PQ and a point on the circle?

Assuming the center of the circle is meant to be (5, 2), we should find both the minimum distance and the maximum distance between the line and the circle. The correct answers should be those two distances and everything in between.

Here's a good rule to know: the minimum distance of anything to a circle should be a straight line from the point to the center of the circle. This line will always, by the way, intercept the circle at a right angle. Anyway, in this problem we can see that drawing a line from the top/bottom of the line to the center of the circle will be longer than if we draw a line from the center of line PQ to the center of the circle. (This is because if we do both, you can see that the line from the top/bottom would be the hypotenuse of a right triangle, while the center of the line would be forming a leg of the same triangle, and thus is shorter.)

Since the circle has a center at (5, 2) and a radius of 1, we know that the left-hand side of the circle is at (4, 2) and a line from point (1, 2) to (4, 2) will have a length of 3. Thus, A is out, B is in, and anything larger than B is potentially in.

What's the maximum length? This is a tougher question. But since we've seen that starting at the top/bottom of line PQ makes the distance larger, let's start there. The easiest way to do this is to imagine that we are measuring the distance from the bottom of line PQ to the closest part of the circle. Then, the farthest part of the circle must be the exact opposite side of the circle. Since the closest side and the farthest side must be exactly 2 apart, (since the diameter is 2), we can just add 2 to this distance.

However, it's not super easy to find the closest part of the circle to point (1, 1), since we don't know exactly where it is. On the other hand, we do know exactly where the center of the circle is: (5, 2). We can find the distance from (1, 1) to (5, 2) by using the Pythagorean formula. The vertical distance of the right triangle formed by these two points is 1, and the horizontal distance is 4, so the Pythagorean formula tells us that:

1^2 + 4^2 = distance^2

so the distance is √17. This takes us to the center of the circle, but the far side must be another 1, since that's just adding the radius. So at this point we have the maximum distance as √17 + 1. What's √17 though? We can estimate it as a tiny bit more than 4, since √16 equals 4. So let's estimate the maximum distance as 5.1. After all, 5 squared is 25, and 17 is far closer to 16 than it is to 25. The next answer choice above 5.1 is 5.5, which is far too high. So F is our maximum at 5.

I would answer everything between B and F, inclusive. Why? If you can imagine the shortest possible line between the middle of line PQ and the left side of the circle, and then imagine smoothly sliding it all the way around the circle to the other side, we'll get an infinite set of distances until we reach the other side, which is a distance of 5. Then if we slide the side of the line that is on line PQ down to the bottom we'll increase the length by just a bit. But since we can slide it continuously we have a range of values.
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